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Full text of "Analytical Mechanics"

258
ANALYTICAL MECHANICS
204. Oblique Impact of a Particle upon a Fixed Plane. Case I. Smooth Contact.  Let vt and vn, Fig. 121, be the components of the velocity along the plane and along the normal, respectively, just before the impact; and let vtf and vnf be the corresponding components just after the impact. Since the plane is smooth, no horizontal forces arise during the impact; hence the horizontal component of the momentum remains constant. Therefore
mvt = wiv/, or               vt = w/.
So far as the vertical component is concerned the impact is direct; therefore
L---k'
FIG. 121.
Denoting by a and ft the angles which the resultant velocity makes with the normal just before and just after the impact we obtain
7} tan a =  S
tan ft =  L-evn
.-.    tan a = e tan p.                     (XIV)
DISCUSSION.  Whefn the contact is perfectly elastic e = 1 ; therefore the angle of incidence equals the angle of reflection as in the case of the reflection of light. In this case the magnitude of the velocity is not changed by the impact, as is to be expected from the conservation of energy.. When the contact is imperfectly elastic the angle of reflection lies between - and the angle of incidence, while the normal component of
<u
the velocity and consequently the magnitude of the total velocity is diminished.   When the contact is perfectly inelastic e = 0, and since a is
not zero ft must be - in order that etan ft may have a finite value.   There-
2i
:ore in this case the particle slides along the plane after the collision.